I only wear diapers in public every once in a while. I get a little nervous, and feel like I'm pushing my luck if I do it too often.
That being said, I find winter is a good time to do it, because you can really layer up, and that helps hide the diaper. I usually wear something more discreet like Attends or Depends, as opposed to all my thick printed diapers. And I have a plain black work shirt onesie, to help keep the diaper in place, and hopefully hide the crinkling a bit. Plus it helps prevent the diaper from sticking out the back or front of your pants. Then I wear my waterproof pants, they're loose fitting, and make swishy noises when I walk, to further help hide the crinkle. Plus they're black, so if they get wet, you can't really tell, and they dry quickly. Then a hoodie to further help cover things.
I've worn around people I know before, and no one seems to have noticed. And generally speaking, most people aren't staring at every random person who walks by, to try and see if they're wearing a diaper. I'm pretty sure it doesn't cross most peoples minds, unless they wear diapers themselves. And even if they did suspect, so what? Maybe you need to wear diapers, and what you choose to wear for "undergarments" really isn't anyone else's business.
As far as the anxiety, I get nervous beforehand, which is why I only do it every once in a while. But once I'm in public, I feel strangely confident. I'm normally very socially anxious, and I'm not a fan of crowds at all. But being in a crowd while wearing a diaper underneath, actually makes me feel pretty confident, which seems completely backwards haha. But I guess I sort of feel like I'm actually doing something I want to do, wearing something I want to wear, instead of bowing to social pressure. I'm not sure if anyone else experiences the same thing, or if I'm alone in that haha.